Alternator or battery or ?

[Bob Horowski]

Not sure what battery maintainer you have but everyone I have owned and seen YES you can connect the eyelets incorrectly to the battery. The battery connectors are simply eyelets that are marked red for positive and black for negative. What you seem to be describing is the other end of the cable which can be connected to alligator clips for a temporary connection or eyelets for a more permanent connection.

 

[Pushrod8Speed]

If alternator is charging and batteries are good and new, sounds like a parasitic draw with key (ignition off). Screen should probably be using ignition on/accessory power only and should be plug and play (no special wiring).
With everything off, check the amperage draw and pull fuses until you find the culprit circuit.
(Edit) forgot to mention it needs a few minutes to go to sleep with nothing on, doors closed, and all key fobs far away.